text: /*                                                      lgammal
   *
   *      Natural logarithm of gamma function
   *
   *
   *
   * SYNOPSIS:
   *
   * long double x, y, lgammal();
   * extern int sgngam;
   *
   * y = lgammal(x);
   *
   *
   *
   * DESCRIPTION:
   *
   * Returns the base e (2.718...) logarithm of the absolute
   * value of the gamma function of the argument.
   * The sign (+1 or -1) of the gamma function is returned in a
   * global (extern) variable named sgngam.
   *
   * The positive domain is partitioned into numerous segments for approximation.
   * For x > 10,
   *   log gamma(x) = (x - 0.5) log(x) - x + log sqrt(2 pi) + 1/x R(1/x^2)
   * Near the minimum at x = x0 = 1.46... the approximation is
   *   log gamma(x0 + z) = log gamma(x0) + z^2 P(z)/Q(z)
   * for small z.
   * Elsewhere between 0 and 10,
   *   log gamma(n + z) = log gamma(n) + z P(z)/Q(z)
   * for various selected n and small z.
   *
   * The cosecant reflection formula is employed for negative arguments.
   *
   *
   *
   * ACCURACY:
   *
   *
   * arithmetic      domain        # trials     peak         rms
   *                                            Relative error:
   *    IEEE         10, 30         100000     3.9e-34     9.8e-35
   *    IEEE          0, 10         100000     3.8e-34     5.3e-35
   *                                            Absolute error:
   *    IEEE         -10, 0         100000     8.0e-34     8.0e-35
   *    IEEE         -30, -10       100000     4.4e-34     1.0e-34
   *    IEEE        -100, 100       100000                 1.0e-34
   *
   * The absolute error criterion is the same as relative error
   * when the function magnitude is greater than one but it is absolute
   * when the magnitude is less than one.
   *
   */
  
  /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
  
      This library is free software; you can redistribute it and/or
      modify it under the terms of the GNU Lesser General Public
      License as published by the Free Software Foundation; either
      version 2.1 of the License, or (at your option) any later version.
  
      This library is distributed in the hope that it will be useful,
      but WITHOUT ANY WARRANTY; without even the implied warranty of
      MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
      Lesser General Public License for more details.
  
      You should have received a copy of the GNU Lesser General Public
      License along with this library; if not, see
      <http://www.gnu.org/licenses/>.  */
  
  #include <math.h>
  #include <math_private.h>
  #include <libc-internal.h>
  #include <float.h>
  
  /* BZ#16347: ldbl-128ibm uses this file as is, however the MAXLGM
     definition overflows for IBM long double.  This directive prevents the
     overflow warnings until IBM long double version is fixed.  */
  static const long double PIL = 3.1415926535897932384626433832795028841972E0L;
  DIAG_PUSH_NEEDS_COMMENT;
  DIAG_IGNORE_NEEDS_COMMENT (4.6, "-Woverflow");
  static const long double MAXLGM = 1.0485738685148938358098967157129705071571E4928L;
  DIAG_POP_NEEDS_COMMENT;
  static const long double one = 1.0L;
  static const long double zero = 0.0L;
  static const long double huge = LDBL_MAX;
  
  /* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2)
     1/x <= 0.0741 (x >= 13.495...)
     Peak relative error 1.5e-36  */
  static const long double ls2pi = 9.1893853320467274178032973640561763986140E-1L;
  #define NRASY 12
  static const long double RASY[NRASY + 1] =
  {
    8.333333333333333333333333333310437112111E-2L,
   -2.777777777777777777777774789556228296902E-3L,
    7.936507936507936507795933938448586499183E-4L,
   -5.952380952380952041799269756378148574045E-4L,
    8.417508417507928904209891117498524452523E-4L,
   -1.917526917481263997778542329739806086290E-3L,
    6.410256381217852504446848671499409919280E-3L,
   -2.955064066900961649768101034477363301626E-2L,
    1.796402955865634243663453415388336954675E-1L,
   -1.391522089007758553455753477688592767741E0L,
    1.326130089598399157988112385013829305510E1L,
   -1.420412699593782497803472576479997819149E2L,
    1.218058922427762808938869872528846787020E3L
  };
  
  
  /* log gamma(x+13) = log gamma(13) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     12.5 <= x+13 <= 13.5
     Peak relative error 1.1e-36  */
  static const long double lgam13a = 1.9987213134765625E1L;
  static const long double lgam13b = 1.3608962611495173623870550785125024484248E-6L;
  #define NRN13 7
  static const long double RN13[NRN13 + 1] =
  {
    8.591478354823578150238226576156275285700E11L,
    2.347931159756482741018258864137297157668E11L,
    2.555408396679352028680662433943000804616E10L,
    1.408581709264464345480765758902967123937E9L,
    4.126759849752613822953004114044451046321E7L,
    6.133298899622688505854211579222889943778E5L,
    3.929248056293651597987893340755876578072E3L,
    6.850783280018706668924952057996075215223E0L
  };
  #define NRD13 6
  static const long double RD13[NRD13 + 1] =
  {
    3.401225382297342302296607039352935541669E11L,
    8.756765276918037910363513243563234551784E10L,
    8.873913342866613213078554180987647243903E9L,
    4.483797255342763263361893016049310017973E8L,
    1.178186288833066430952276702931512870676E7L,
    1.519928623743264797939103740132278337476E5L,
    7.989298844938119228411117593338850892311E2L
   /* 1.0E0L */
  };
  
  
  /* log gamma(x+12) = log gamma(12) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     11.5 <= x+12 <= 12.5
     Peak relative error 4.1e-36  */
  static const long double lgam12a = 1.75023040771484375E1L;
  static const long double lgam12b = 3.7687254483392876529072161996717039575982E-6L;
  #define NRN12 7
  static const long double RN12[NRN12 + 1] =
  {
    4.709859662695606986110997348630997559137E11L,
    1.398713878079497115037857470168777995230E11L,
    1.654654931821564315970930093932954900867E10L,
    9.916279414876676861193649489207282144036E8L,
    3.159604070526036074112008954113411389879E7L,
    5.109099197547205212294747623977502492861E5L,
    3.563054878276102790183396740969279826988E3L,
    6.769610657004672719224614163196946862747E0L
  };
  #define NRD12 6
  static const long double RD12[NRD12 + 1] =
  {
    1.928167007860968063912467318985802726613E11L,
    5.383198282277806237247492369072266389233E10L,
    5.915693215338294477444809323037871058363E9L,
    3.241438287570196713148310560147925781342E8L,
    9.236680081763754597872713592701048455890E6L,
    1.292246897881650919242713651166596478850E5L,
    7.366532445427159272584194816076600211171E2L
   /* 1.0E0L */
  };
  
  
  /* log gamma(x+11) = log gamma(11) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     10.5 <= x+11 <= 11.5
     Peak relative error 1.8e-35  */
  static const long double lgam11a = 1.5104400634765625E1L;
  static const long double lgam11b = 1.1938309890295225709329251070371882250744E-5L;
  #define NRN11 7
  static const long double RN11[NRN11 + 1] =
  {
    2.446960438029415837384622675816736622795E11L,
    7.955444974446413315803799763901729640350E10L,
    1.030555327949159293591618473447420338444E10L,
    6.765022131195302709153994345470493334946E8L,
    2.361892792609204855279723576041468347494E7L,
    4.186623629779479136428005806072176490125E5L,
    3.202506022088912768601325534149383594049E3L,
    6.681356101133728289358838690666225691363E0L
  };
  #define NRD11 6
  static const long double RD11[NRD11 + 1] =
  {
    1.040483786179428590683912396379079477432E11L,
    3.172251138489229497223696648369823779729E10L,
    3.806961885984850433709295832245848084614E9L,
    2.278070344022934913730015420611609620171E8L,
    7.089478198662651683977290023829391596481E6L,
    1.083246385105903533237139380509590158658E5L,
    6.744420991491385145885727942219463243597E2L
   /* 1.0E0L */
  };
  
  
  /* log gamma(x+10) = log gamma(10) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     9.5 <= x+10 <= 10.5
     Peak relative error 5.4e-37  */
  static const long double lgam10a = 1.280181884765625E1L;
  static const long double lgam10b = 8.6324252196112077178745667061642811492557E-6L;
  #define NRN10 7
  static const long double RN10[NRN10 + 1] =
  {
    -1.239059737177249934158597996648808363783E14L,
    -4.725899566371458992365624673357356908719E13L,
    -7.283906268647083312042059082837754850808E12L,
    -5.802855515464011422171165179767478794637E11L,
    -2.532349691157548788382820303182745897298E10L,
    -5.884260178023777312587193693477072061820E8L,
    -6.437774864512125749845840472131829114906E6L,
    -2.350975266781548931856017239843273049384E4L
  };
  #define NRD10 7
  static const long double RD10[NRD10 + 1] =
  {
    -5.502645997581822567468347817182347679552E13L,
    -1.970266640239849804162284805400136473801E13L,
    -2.819677689615038489384974042561531409392E12L,
    -2.056105863694742752589691183194061265094E11L,
    -8.053670086493258693186307810815819662078E9L,
    -1.632090155573373286153427982504851867131E8L,
    -1.483575879240631280658077826889223634921E6L,
    -4.002806669713232271615885826373550502510E3L
   /* 1.0E0L */
  };
  
  
  /* log gamma(x+9) = log gamma(9) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     8.5 <= x+9 <= 9.5
     Peak relative error 3.6e-36  */
  static const long double lgam9a = 1.06045989990234375E1L;
  static const long double lgam9b = 3.9037218127284172274007216547549861681400E-6L;
  #define NRN9 7
  static const long double RN9[NRN9 + 1] =
  {
    -4.936332264202687973364500998984608306189E13L,
    -2.101372682623700967335206138517766274855E13L,
    -3.615893404644823888655732817505129444195E12L,
    -3.217104993800878891194322691860075472926E11L,
    -1.568465330337375725685439173603032921399E10L,
    -4.073317518162025744377629219101510217761E8L,
    -4.983232096406156139324846656819246974500E6L,
    -2.036280038903695980912289722995505277253E4L
  };
  #define NRD9 7
  static const long double RD9[NRD9 + 1] =
  {
    -2.306006080437656357167128541231915480393E13L,
    -9.183606842453274924895648863832233799950E12L,
    -1.461857965935942962087907301194381010380E12L,
    -1.185728254682789754150068652663124298303E11L,
    -5.166285094703468567389566085480783070037E9L,
    -1.164573656694603024184768200787835094317E8L,
    -1.177343939483908678474886454113163527909E6L,
    -3.529391059783109732159524500029157638736E3L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+8) = log gamma(8) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     7.5 <= x+8 <= 8.5
     Peak relative error 2.4e-37  */
  static const long double lgam8a = 8.525146484375E0L;
  static const long double lgam8b = 1.4876690414300165531036347125050759667737E-5L;
  #define NRN8 8
  static const long double RN8[NRN8 + 1] =
  {
    6.600775438203423546565361176829139703289E11L,
    3.406361267593790705240802723914281025800E11L,
    7.222460928505293914746983300555538432830E10L,
    8.102984106025088123058747466840656458342E9L,
    5.157620015986282905232150979772409345927E8L,
    1.851445288272645829028129389609068641517E7L,
    3.489261702223124354745894067468953756656E5L,
    2.892095396706665774434217489775617756014E3L,
    6.596977510622195827183948478627058738034E0L
  };
  #define NRD8 7
  static const long double RD8[NRD8 + 1] =
  {
    3.274776546520735414638114828622673016920E11L,
    1.581811207929065544043963828487733970107E11L,
    3.108725655667825188135393076860104546416E10L,
    3.193055010502912617128480163681842165730E9L,
    1.830871482669835106357529710116211541839E8L,
    5.790862854275238129848491555068073485086E6L,
    9.305213264307921522842678835618803553589E4L,
    6.216974105861848386918949336819572333622E2L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+7) = log gamma(7) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     6.5 <= x+7 <= 7.5
     Peak relative error 3.2e-36  */
  static const long double lgam7a = 6.5792388916015625E0L;
  static const long double lgam7b = 1.2320408538495060178292903945321122583007E-5L;
  #define NRN7 8
  static const long double RN7[NRN7 + 1] =
  {
    2.065019306969459407636744543358209942213E11L,
    1.226919919023736909889724951708796532847E11L,
    2.996157990374348596472241776917953749106E10L,
    3.873001919306801037344727168434909521030E9L,
    2.841575255593761593270885753992732145094E8L,
    1.176342515359431913664715324652399565551E7L,
    2.558097039684188723597519300356028511547E5L,
    2.448525238332609439023786244782810774702E3L,
    6.460280377802030953041566617300902020435E0L
  };
  #define NRD7 7
  static const long double RD7[NRD7 + 1] =
  {
    1.102646614598516998880874785339049304483E11L,
    6.099297512712715445879759589407189290040E10L,
    1.372898136289611312713283201112060238351E10L,
    1.615306270420293159907951633566635172343E9L,
    1.061114435798489135996614242842561967459E8L,
    3.845638971184305248268608902030718674691E6L,
    7.081730675423444975703917836972720495507E4L,
    5.423122582741398226693137276201344096370E2L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+6) = log gamma(6) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     5.5 <= x+6 <= 6.5
     Peak relative error 6.2e-37  */
  static const long double lgam6a = 4.7874908447265625E0L;
  static const long double lgam6b = 8.9805548349424770093452324304839959231517E-7L;
  #define NRN6 8
  static const long double RN6[NRN6 + 1] =
  {
    -3.538412754670746879119162116819571823643E13L,
    -2.613432593406849155765698121483394257148E13L,
    -8.020670732770461579558867891923784753062E12L,
    -1.322227822931250045347591780332435433420E12L,
    -1.262809382777272476572558806855377129513E11L,
    -7.015006277027660872284922325741197022467E9L,
    -2.149320689089020841076532186783055727299E8L,
    -3.167210585700002703820077565539658995316E6L,
    -1.576834867378554185210279285358586385266E4L
  };
  #define NRD6 8
  static const long double RD6[NRD6 + 1] =
  {
    -2.073955870771283609792355579558899389085E13L,
    -1.421592856111673959642750863283919318175E13L,
    -4.012134994918353924219048850264207074949E12L,
    -6.013361045800992316498238470888523722431E11L,
    -5.145382510136622274784240527039643430628E10L,
    -2.510575820013409711678540476918249524123E9L,
    -6.564058379709759600836745035871373240904E7L,
    -7.861511116647120540275354855221373571536E5L,
    -2.821943442729620524365661338459579270561E3L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+5) = log gamma(5) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     4.5 <= x+5 <= 5.5
     Peak relative error 3.4e-37  */
  static const long double lgam5a = 3.17803955078125E0L;
  static const long double lgam5b = 1.4279566695619646941601297055408873990961E-5L;
  #define NRN5 9
  static const long double RN5[NRN5 + 1] =
  {
    2.010952885441805899580403215533972172098E11L,
    1.916132681242540921354921906708215338584E11L,
    7.679102403710581712903937970163206882492E10L,
    1.680514903671382470108010973615268125169E10L,
    2.181011222911537259440775283277711588410E9L,
    1.705361119398837808244780667539728356096E8L,
    7.792391565652481864976147945997033946360E6L,
    1.910741381027985291688667214472560023819E5L,
    2.088138241893612679762260077783794329559E3L,
    6.330318119566998299106803922739066556550E0L
  };
  #define NRD5 8
  static const long double RD5[NRD5 + 1] =
  {
    1.335189758138651840605141370223112376176E11L,
    1.174130445739492885895466097516530211283E11L,
    4.308006619274572338118732154886328519910E10L,
    8.547402888692578655814445003283720677468E9L,
    9.934628078575618309542580800421370730906E8L,
    6.847107420092173812998096295422311820672E7L,
    2.698552646016599923609773122139463150403E6L,
    5.526516251532464176412113632726150253215E4L,
    4.772343321713697385780533022595450486932E2L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+4) = log gamma(4) +  x P(x)/Q(x)
     -0.5 <= x <= 0.5
     3.5 <= x+4 <= 4.5
     Peak relative error 6.7e-37  */
  static const long double lgam4a = 1.791748046875E0L;
  static const long double lgam4b = 1.1422353055000812477358380702272722990692E-5L;
  #define NRN4 9
  static const long double RN4[NRN4 + 1] =
  {
    -1.026583408246155508572442242188887829208E13L,
    -1.306476685384622809290193031208776258809E13L,
    -7.051088602207062164232806511992978915508E12L,
    -2.100849457735620004967624442027793656108E12L,
    -3.767473790774546963588549871673843260569E11L,
    -4.156387497364909963498394522336575984206E10L,
    -2.764021460668011732047778992419118757746E9L,
    -1.036617204107109779944986471142938641399E8L,
    -1.895730886640349026257780896972598305443E6L,
    -1.180509051468390914200720003907727988201E4L
  };
  #define NRD4 9
  static const long double RD4[NRD4 + 1] =
  {
    -8.172669122056002077809119378047536240889E12L,
    -9.477592426087986751343695251801814226960E12L,
    -4.629448850139318158743900253637212801682E12L,
    -1.237965465892012573255370078308035272942E12L,
    -1.971624313506929845158062177061297598956E11L,
    -1.905434843346570533229942397763361493610E10L,
    -1.089409357680461419743730978512856675984E9L,
    -3.416703082301143192939774401370222822430E7L,
    -4.981791914177103793218433195857635265295E5L,
    -2.192507743896742751483055798411231453733E3L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+3) = log gamma(3) +  x P(x)/Q(x)
     -0.25 <= x <= 0.5
     2.75 <= x+3 <= 3.5
     Peak relative error 6.0e-37  */
  static const long double lgam3a = 6.93145751953125E-1L;
  static const long double lgam3b = 1.4286068203094172321214581765680755001344E-6L;
  
  #define NRN3 9
  static const long double RN3[NRN3 + 1] =
  {
    -4.813901815114776281494823863935820876670E11L,
    -8.425592975288250400493910291066881992620E11L,
    -6.228685507402467503655405482985516909157E11L,
    -2.531972054436786351403749276956707260499E11L,
    -6.170200796658926701311867484296426831687E10L,
    -9.211477458528156048231908798456365081135E9L,
    -8.251806236175037114064561038908691305583E8L,
    -4.147886355917831049939930101151160447495E7L,
    -1.010851868928346082547075956946476932162E6L,
    -8.333374463411801009783402800801201603736E3L
  };
  #define NRD3 9
  static const long double RD3[NRD3 + 1] =
  {
    -5.216713843111675050627304523368029262450E11L,
    -8.014292925418308759369583419234079164391E11L,
    -5.180106858220030014546267824392678611990E11L,
    -1.830406975497439003897734969120997840011E11L,
    -3.845274631904879621945745960119924118925E10L,
    -4.891033385370523863288908070309417710903E9L,
    -3.670172254411328640353855768698287474282E8L,
    -1.505316381525727713026364396635522516989E7L,
    -2.856327162923716881454613540575964890347E5L,
    -1.622140448015769906847567212766206894547E3L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+2.5) = log gamma(2.5) +  x P(x)/Q(x)
     -0.125 <= x <= 0.25
     2.375 <= x+2.5 <= 2.75  */
  static const long double lgam2r5a = 2.8466796875E-1L;
  static const long double lgam2r5b = 1.4901722919159632494669682701924320137696E-5L;
  #define NRN2r5 8
  static const long double RN2r5[NRN2r5 + 1] =
  {
    -4.676454313888335499356699817678862233205E9L,
    -9.361888347911187924389905984624216340639E9L,
    -7.695353600835685037920815799526540237703E9L,
    -3.364370100981509060441853085968900734521E9L,
    -8.449902011848163568670361316804900559863E8L,
    -1.225249050950801905108001246436783022179E8L,
    -9.732972931077110161639900388121650470926E6L,
    -3.695711763932153505623248207576425983573E5L,
    -4.717341584067827676530426007495274711306E3L
  };
  #define NRD2r5 8
  static const long double RD2r5[NRD2r5 + 1] =
  {
    -6.650657966618993679456019224416926875619E9L,
    -1.099511409330635807899718829033488771623E10L,
    -7.482546968307837168164311101447116903148E9L,
    -2.702967190056506495988922973755870557217E9L,
    -5.570008176482922704972943389590409280950E8L,
    -6.536934032192792470926310043166993233231E7L,
    -4.101991193844953082400035444146067511725E6L,
    -1.174082735875715802334430481065526664020E5L,
    -9.932840389994157592102947657277692978511E2L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+2) = x P(x)/Q(x)
     -0.125 <= x <= +0.375
     1.875 <= x+2 <= 2.375
     Peak relative error 4.6e-36  */
  #define NRN2 9
  static const long double RN2[NRN2 + 1] =
  {
    -3.716661929737318153526921358113793421524E9L,
    -1.138816715030710406922819131397532331321E10L,
    -1.421017419363526524544402598734013569950E10L,
    -9.510432842542519665483662502132010331451E9L,
    -3.747528562099410197957514973274474767329E9L,
    -8.923565763363912474488712255317033616626E8L,
    -1.261396653700237624185350402781338231697E8L,
    -9.918402520255661797735331317081425749014E6L,
    -3.753996255897143855113273724233104768831E5L,
    -4.778761333044147141559311805999540765612E3L
  };
  #define NRD2 9
  static const long double RD2[NRD2 + 1] =
  {
    -8.790916836764308497770359421351673950111E9L,
    -2.023108608053212516399197678553737477486E10L,
    -1.958067901852022239294231785363504458367E10L,
    -1.035515043621003101254252481625188704529E10L,
    -3.253884432621336737640841276619272224476E9L,
    -6.186383531162456814954947669274235815544E8L,
    -6.932557847749518463038934953605969951466E7L,
    -4.240731768287359608773351626528479703758E6L,
    -1.197343995089189188078944689846348116630E5L,
    -1.004622911670588064824904487064114090920E3L
  /* 1.0E0 */
  };
  
  
  /* log gamma(x+1.75) = log gamma(1.75) +  x P(x)/Q(x)
     -0.125 <= x <= +0.125
     1.625 <= x+1.75 <= 1.875
     Peak relative error 9.2e-37 */
  static const long double lgam1r75a = -8.441162109375E-2L;
  static const long double lgam1r75b = 1.0500073264444042213965868602268256157604E-5L;
  #define NRN1r75 8
  static const long double RN1r75[NRN1r75 + 1] =
  {
    -5.221061693929833937710891646275798251513E7L,
    -2.052466337474314812817883030472496436993E8L,
    -2.952718275974940270675670705084125640069E8L,
    -2.132294039648116684922965964126389017840E8L,
    -8.554103077186505960591321962207519908489E7L,
    -1.940250901348870867323943119132071960050E7L,
    -2.379394147112756860769336400290402208435E6L,
    -1.384060879999526222029386539622255797389E5L,
    -2.698453601378319296159355612094598695530E3L
  };
  #define NRD1r75 8
  static const long double RD1r75[NRD1r75 + 1] =
  {
    -2.109754689501705828789976311354395393605E8L,
    -5.036651829232895725959911504899241062286E8L,
    -4.954234699418689764943486770327295098084E8L,
    -2.589558042412676610775157783898195339410E8L,
    -7.731476117252958268044969614034776883031E7L,
    -1.316721702252481296030801191240867486965E7L,
    -1.201296501404876774861190604303728810836E6L,
    -5.007966406976106636109459072523610273928E4L,
    -6.155817990560743422008969155276229018209E2L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+x0) = y0 +  x^2 P(x)/Q(x)
     -0.0867 <= x <= +0.1634
     1.374932... <= x+x0 <= 1.625032...
     Peak relative error 4.0e-36  */
  static const long double x0a = 1.4616241455078125L;
  static const long double x0b = 7.9994605498412626595423257213002588621246E-6L;
  static const long double y0a = -1.21490478515625E-1L;
  static const long double y0b = 4.1879797753919044854428223084178486438269E-6L;
  #define NRN1r5 8
  static const long double RN1r5[NRN1r5 + 1] =
  {
    6.827103657233705798067415468881313128066E5L,
    1.910041815932269464714909706705242148108E6L,
    2.194344176925978377083808566251427771951E6L,
    1.332921400100891472195055269688876427962E6L,
    4.589080973377307211815655093824787123508E5L,
    8.900334161263456942727083580232613796141E4L,
    9.053840838306019753209127312097612455236E3L,
    4.053367147553353374151852319743594873771E2L,
    5.040631576303952022968949605613514584950E0L
  };
  #define NRD1r5 8
  static const long double RD1r5[NRD1r5 + 1] =
  {
    1.411036368843183477558773688484699813355E6L,
    4.378121767236251950226362443134306184849E6L,
    5.682322855631723455425929877581697918168E6L,
    3.999065731556977782435009349967042222375E6L,
    1.653651390456781293163585493620758410333E6L,
    4.067774359067489605179546964969435858311E5L,
    5.741463295366557346748361781768833633256E4L,
    4.226404539738182992856094681115746692030E3L,
    1.316980975410327975566999780608618774469E2L,
    /* 1.0E0L */
  };
  
  
  /* log gamma(x+1.25) = log gamma(1.25) +  x P(x)/Q(x)
     -.125 <= x <= +.125
     1.125 <= x+1.25 <= 1.375
     Peak relative error = 4.9e-36 */
  static const long double lgam1r25a = -9.82818603515625E-2L;
  static const long double lgam1r25b = 1.0023929749338536146197303364159774377296E-5L;
  #define NRN1r25 9
  static const long double RN1r25[NRN1r25 + 1] =
  {
    -9.054787275312026472896002240379580536760E4L,
    -8.685076892989927640126560802094680794471E4L,
    2.797898965448019916967849727279076547109E5L,
    6.175520827134342734546868356396008898299E5L,
    5.179626599589134831538516906517372619641E5L,
    2.253076616239043944538380039205558242161E5L,
    5.312653119599957228630544772499197307195E4L,
    6.434329437514083776052669599834938898255E3L,
    3.385414416983114598582554037612347549220E2L,
    4.907821957946273805080625052510832015792E0L
  };
  #define NRD1r25 8
  static const long double RD1r25[NRD1r25 + 1] =
  {
    3.980939377333448005389084785896660309000E5L,
    1.429634893085231519692365775184490465542E6L,
    2.145438946455476062850151428438668234336E6L,
    1.743786661358280837020848127465970357893E6L,
    8.316364251289743923178092656080441655273E5L,
    2.355732939106812496699621491135458324294E5L,
    3.822267399625696880571810137601310855419E4L,
    3.228463206479133236028576845538387620856E3L,
    1.152133170470059555646301189220117965514E2L
    /* 1.0E0L */
  };
  
  
  /* log gamma(x + 1) = x P(x)/Q(x)
     0.0 <= x <= +0.125
     1.0 <= x+1 <= 1.125
     Peak relative error 1.1e-35  */
  #define NRN1 8
  static const long double RN1[NRN1 + 1] =
  {
    -9.987560186094800756471055681088744738818E3L,
    -2.506039379419574361949680225279376329742E4L,
    -1.386770737662176516403363873617457652991E4L,
    1.439445846078103202928677244188837130744E4L,
    2.159612048879650471489449668295139990693E4L,
    1.047439813638144485276023138173676047079E4L,
    2.250316398054332592560412486630769139961E3L,
    1.958510425467720733041971651126443864041E2L,
    4.516830313569454663374271993200291219855E0L
  };
  #define NRD1 7
  static const long double RD1[NRD1 + 1] =
  {
    1.730299573175751778863269333703788214547E4L,
    6.807080914851328611903744668028014678148E4L,
    1.090071629101496938655806063184092302439E5L,
    9.124354356415154289343303999616003884080E4L,
    4.262071638655772404431164427024003253954E4L,
    1.096981664067373953673982635805821283581E4L,
    1.431229503796575892151252708527595787588E3L,
    7.734110684303689320830401788262295992921E1L
   /* 1.0E0 */
  };
  
  
  /* log gamma(x + 1) = x P(x)/Q(x)
     -0.125 <= x <= 0
     0.875 <= x+1 <= 1.0
     Peak relative error 7.0e-37  */
  #define NRNr9 8
  static const long double RNr9[NRNr9 + 1] =
  {
    4.441379198241760069548832023257571176884E5L,
    1.273072988367176540909122090089580368732E6L,
    9.732422305818501557502584486510048387724E5L,
    -5.040539994443998275271644292272870348684E5L,
    -1.208719055525609446357448132109723786736E6L,
    -7.434275365370936547146540554419058907156E5L,
    -2.075642969983377738209203358199008185741E5L,
    -2.565534860781128618589288075109372218042E4L,
    -1.032901669542994124131223797515913955938E3L,
  };
  #define NRDr9 8
  static const long double RDr9[NRDr9 + 1] =
  {
    -7.694488331323118759486182246005193998007E5L,
    -3.301918855321234414232308938454112213751E6L,
    -5.856830900232338906742924836032279404702E6L,
    -5.540672519616151584486240871424021377540E6L,
    -3.006530901041386626148342989181721176919E6L,
    -9.350378280513062139466966374330795935163E5L,
    -1.566179100031063346901755685375732739511E5L,
    -1.205016539620260779274902967231510804992E4L,
    -2.724583156305709733221564484006088794284E2L
  /* 1.0E0 */
  };
  
  
  /* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
  
  static long double
  neval (long double x, const long double *p, int n)
  {
    long double y;
  
    p += n;
    y = *p--;
    do
      {
        y = y * x + *p--;
      }
    while (--n > 0);
    return y;
  }
  
  
  /* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
  
  static long double
  deval (long double x, const long double *p, int n)
  {
    long double y;
  
    p += n;
    y = x + *p--;
    do
      {
        y = y * x + *p--;
      }
    while (--n > 0);
    return y;
  }
  
  
  long double
  __ieee754_lgammal_r (long double x, int *signgamp)
  {
    long double p, q, w, z, nx;
    int i, nn;
  
    *signgamp = 1;
  
    if (! __finitel (x))
      return x * x;
  
    if (x == 0.0L)
      {
        if (__signbitl (x))
  	*signgamp = -1;
      }
  
    if (x < 0.0L)
      {
        q = -x;
        p = __floorl (q);
        if (p == q)
  	return (one / (p - p));
        i = p;
        if ((i & 1) == 0)
  	*signgamp = -1;
        else
  	*signgamp = 1;
        if (q < 0x1p-120L)
  	return -__logl (q);
        z = q - p;
        if (z > 0.5L)
  	{
  	  p += 1.0L;
  	  z = p - q;
  	}
        z = q * __sinl (PIL * z);
        w = __ieee754_lgammal_r (q, &i);
        z = __logl (PIL / z) - w;
        return (z);
      }
  
    if (x < 13.5L)
      {
        p = 0.0L;
        nx = __floorl (x + 0.5L);
        nn = nx;
        switch (nn)
  	{
  	case 0:
  	  /* log gamma (x + 1) = log(x) + log gamma(x) */
  	  if (x < 0x1p-120L)
  	    return -__logl (x);
  	  else if (x <= 0.125)
  	    {
  	      p = x * neval (x, RN1, NRN1) / deval (x, RD1, NRD1);
  	    }
  	  else if (x <= 0.375)
  	    {
  	      z = x - 0.25L;
  	      p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
  	      p += lgam1r25b;
  	      p += lgam1r25a;
  	    }
  	  else if (x <= 0.625)
  	    {
  	      z = x + (1.0L - x0a);
  	      z = z - x0b;
  	      p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
  	      p = p * z * z;
  	      p = p + y0b;
  	      p = p + y0a;
  	    }
  	  else if (x <= 0.875)
  	    {
  	      z = x - 0.75L;
  	      p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
  	      p += lgam1r75b;
  	      p += lgam1r75a;
  	    }
  	  else
  	    {
  	      z = x - 1.0L;
  	      p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
  	    }
  	  p = p - __logl (x);
  	  break;
  
  	case 1:
  	  if (x < 0.875L)
  	    {
  	      if (x <= 0.625)
  		{
  		  z = x + (1.0L - x0a);
  		  z = z - x0b;
  		  p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
  		  p = p * z * z;
  		  p = p + y0b;
  		  p = p + y0a;
  		}
  	      else if (x <= 0.875)
  		{
  		  z = x - 0.75L;
  		  p = z * neval (z, RN1r75, NRN1r75)
  			/ deval (z, RD1r75, NRD1r75);
  		  p += lgam1r75b;
  		  p += lgam1r75a;
  		}
  	      else
  		{
  		  z = x - 1.0L;
  		  p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
  		}
  	      p = p - __logl (x);
  	    }
  	  else if (x < 1.0L)
  	    {
  	      z = x - 1.0L;
  	      p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9);
  	    }
  	  else if (x == 1.0L)
  	    p = 0.0L;
  	  else if (x <= 1.125L)
  	    {
  	      z = x - 1.0L;
  	      p = z * neval (z, RN1, NRN1) / deval (z, RD1, NRD1);
  	    }
  	  else if (x <= 1.375)
  	    {
  	      z = x - 1.25L;
  	      p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
  	      p += lgam1r25b;
  	      p += lgam1r25a;
  	    }
  	  else
  	    {
  	      /* 1.375 <= x+x0 <= 1.625 */
  	      z = x - x0a;
  	      z = z - x0b;
  	      p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
  	      p = p * z * z;
  	      p = p + y0b;
  	      p = p + y0a;
  	    }
  	  break;
  
  	case 2:
  	  if (x < 1.625L)
  	    {
  	      z = x - x0a;
  	      z = z - x0b;
  	      p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
  	      p = p * z * z;
  	      p = p + y0b;
  	      p = p + y0a;
  	    }
  	  else if (x < 1.875L)
  	    {
  	      z = x - 1.75L;
  	      p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
  	      p += lgam1r75b;
  	      p += lgam1r75a;
  	    }
  	  else if (x == 2.0L)
  	    p = 0.0L;
  	  else if (x < 2.375L)
  	    {
  	      z = x - 2.0L;
  	      p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
  	    }
  	  else
  	    {
  	      z = x - 2.5L;
  	      p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
  	      p += lgam2r5b;
  	      p += lgam2r5a;
  	    }
  	  break;
  
  	case 3:
  	  if (x < 2.75)
  	    {
  	      z = x - 2.5L;
  	      p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
  	      p += lgam2r5b;
  	      p += lgam2r5a;
  	    }
  	  else
  	    {
  	      z = x - 3.0L;
  	      p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3);
  	      p += lgam3b;
  	      p += lgam3a;
  	    }
  	  break;
  
  	case 4:
  	  z = x - 4.0L;
  	  p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4);
  	  p += lgam4b;
  	  p += lgam4a;
  	  break;
  
  	case 5:
  	  z = x - 5.0L;
  	  p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5);
  	  p += lgam5b;
  	  p += lgam5a;
  	  break;
  
  	case 6:
  	  z = x - 6.0L;
  	  p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6);
  	  p += lgam6b;
  	  p += lgam6a;
  	  break;
  
  	case 7:
  	  z = x - 7.0L;
  	  p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7);
  	  p += lgam7b;
  	  p += lgam7a;
  	  break;
  
  	case 8:
  	  z = x - 8.0L;
  	  p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8);
  	  p += lgam8b;
  	  p += lgam8a;
  	  break;
  
  	case 9:
  	  z = x - 9.0L;
  	  p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9);
  	  p += lgam9b;
  	  p += lgam9a;
  	  break;
  
  	case 10:
  	  z = x - 10.0L;
  	  p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10);
  	  p += lgam10b;
  	  p += lgam10a;
  	  break;
  
  	case 11:
  	  z = x - 11.0L;
  	  p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11);
  	  p += lgam11b;
  	  p += lgam11a;
  	  break;
  
  	case 12:
  	  z = x - 12.0L;
  	  p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12);
  	  p += lgam12b;
  	  p += lgam12a;
  	  break;
  
  	case 13:
  	  z = x - 13.0L;
  	  p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13);
  	  p += lgam13b;
  	  p += lgam13a;
  	  break;
  	}
        return p;
      }
  
    if (x > MAXLGM)
      return (*signgamp * huge * huge);
  
    q = ls2pi - x;
    q = (x - 0.5L) * __logl (x) + q;
    if (x > 1.0e18L)
      return (q);
  
    p = 1.0L / (x * x);
    q += neval (p, RASY, NRASY) / x;
    return (q);
  }
  strong_alias (__ieee754_lgammal_r, __lgammal_r_finite)
  